One confusion I wrote down in advance was “I still don’t quite know how to predict that there will not be a simple mathematical apparatus that explains something. Why the motion of the planets, why the game of chance, why not the color of houses in England or the number of hairs on a man’s head?”
I think the main thing I’d look for is an unusual amount of regularity. This comes in two types:
Natural regularity: unusual ‘spherical cow’ type situations like the movement of the planets. Things that are somehow isolated, or where some particular effect strongly dominates, so that only a few variables are needed
Artificial regularity: a lot of the regularity we see around us is there because people engineered it. Dice and coins are good examples. Can’t remember details but I think there’s some interesting stuff on the history of dice, e.g. this link says that ‘Only in the middle of the 15th century did it become standard to use symmetric cubes’. I think it would be hard to invent probability theory when gambling with irregularly shaped lumps.
There doesn’t seem to be any particularly obvious regularity to house colours or number of hairs, they just look like your standard-issue messy situations that don’t tell you much.
This is exactly the question that John Wentworth is trying to answer with his abstraction hypothesis framework.
Also related to Jaynes proof that probability of a fair coin coming up heads is 1⁄2.
As to being able to discern between different theories. Partly you are right that it can be hard during a scientific controversy and it involves a lot of judgement calls.
On the other hand, it can be hard for layman to appreciate how ‘rigid’ good mathematical models are.
Newton didn’t just observe that apples fall to the ground but he posited a series of elegant laws and was able to calculate very nonobvious results.
The entire theory is quite large and intricate—and there are many quantitive tests one can do and that have been done.
I think the main thing I’d look for is an unusual amount of regularity. This comes in two types:
Natural regularity: unusual ‘spherical cow’ type situations like the movement of the planets. Things that are somehow isolated, or where some particular effect strongly dominates, so that only a few variables are needed
Artificial regularity: a lot of the regularity we see around us is there because people engineered it. Dice and coins are good examples. Can’t remember details but I think there’s some interesting stuff on the history of dice, e.g. this link says that ‘Only in the middle of the 15th century did it become standard to use symmetric cubes’. I think it would be hard to invent probability theory when gambling with irregularly shaped lumps.
There doesn’t seem to be any particularly obvious regularity to house colours or number of hairs, they just look like your standard-issue messy situations that don’t tell you much.
Haha! Those poor people. All of my intuitions about probabilities would have been terribly broken in those times.
This is exactly the question that John Wentworth is trying to answer with his abstraction hypothesis framework. Also related to Jaynes proof that probability of a fair coin coming up heads is 1⁄2.
As to being able to discern between different theories. Partly you are right that it can be hard during a scientific controversy and it involves a lot of judgement calls. On the other hand, it can be hard for layman to appreciate how ‘rigid’ good mathematical models are. Newton didn’t just observe that apples fall to the ground but he posited a series of elegant laws and was able to calculate very nonobvious results. The entire theory is quite large and intricate—and there are many quantitive tests one can do and that have been done.